System, method, and computer program product for importance sampling of partitioned domains

ABSTRACT

A system, method, and computer program product are provided for importance sampling of partitioned domains. In operation, a domain is partitioned into a plurality of sets. Additionally, a probability is assigned to each of the plurality of sets. Furthermore, samples are generated from the plurality of sets, the samples being generated according to the probability of a corresponding set.

FIELD OF THE INVENTION

The present invention relates to sampling a domain, and moreparticularly to importance sampling of partitioned domains.

BACKGROUND

Importance sampling of partitioned domains, where a probability is knownfor each set of the partition, is used for Monte Carlo and quasi-MonteCarlo methods. Monte Carlo methods are a class of algorithms that relyon repeated random sampling to compute their results. Monte Carlomethods are often used when simulating physical and mathematicalsystems. Quasi-Monte Carlo methods are similar to Monte Carlo methods,however, instead deterministic samples are used, which can result in animproved convergence.

Quite often, a function is at least in part defined by some discreteenergy density, such as a textured light source in computer graphics. Todate, the sampling speed for importance sampling of partitioned (ordiscretized) domains has not been fully optimized. There is thus a needfor addressing these and/or other issues.

SUMMARY

A system, method, and computer program product are provided forimportance sampling of partitioned domains. In operation, a domain ispartitioned into a plurality of sets. Additionally, a probability isassigned to each of the plurality of sets. Furthermore, samples aregenerated from the plurality of sets, the samples being generatedaccording to the probability of a corresponding set.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a method for importance sampling of partitioned domains, inaccordance with one embodiment.

FIG. 2 shows a method for enumerating samples using a Huffman tree, inaccordance with another embodiment.

FIG. 3 shows a method for constructing kd-trees with low expectedtraversal depth and importance sampling for such kd-trees, in accordancewith another embodiment.

FIG. 4 shows a probability tree, where inner nodes hold the sum of theprobabilities of their children, in accordance with one embodiment.

FIG. 5 shows a decision tree corresponding to FIG. 4 that may be usedfor sample warping, in accordance with one embodiment.

FIG. 6 shows a tree of sample intervals for each node corresponding toFIG. 5, when 1d-sample warping is used, in accordance with oneembodiment.

FIG. 7 shows a table illustrating statistics for different probes, inaccordance with one embodiment.

FIG. 8 illustrates an exemplary system in which the various architectureand/or functionality of the various previous embodiments may beimplemented.

DETAILED DESCRIPTION

FIG. 1 shows a method 100 for importance sampling of partitioneddomains, in accordance with one embodiment. As shown, a partitioneddomain including a plurality of sets is generated. See operation 102. Inone embodiment, the partitioned domain may be generated by partitioninga domain into a plurality of sets.

In the context of the present description, a partitioned domain refersto any partitioned object or group of partitioned objects. In oneembodiment, the partitioned domain may be an arbitrary-dimensionalpartitioned domain. Additionally, the partitioned objects may be createdby partitioning geometric objects.

In the context of the present description, partitions includevoxelizations, where the sets correspond to volume elements (i.e.voxels) that represent a value on a grid in a space of arbitrarydimension. For example, each voxel may be a quantum unit of volume thathas a numeric value (or values) associated with it that represents somemeasurable properties, independent variables, or attribute of an objector phenomenon. Still yet, any voxel may include a voxel or a group ofvoxels.

Once the partitioned domain including a plurality of sets is generated,a probability is assigned to each of the plurality of sets. Seeoperation 104. Furthermore, samples are generated inside the pluralityof sets, the samples being generated according to the probability of thecorresponding set. See operation 106.

In one embodiment, the samples may be generated utilizing a data tree.For example, in one embodiment, the data tree may include a Huffmantree. In this case, the Huffman tree may be constructed over thepartitioned domain (e.g. utilizing the probability assigned to each ofthe plurality of sets, etc.).

As an option, the Huffman tree may be traversed down to a single domaincell. In one embodiment, the traversing down may be performed utilizingsample warping. In another embodiment, the traversing down may beperformed by generating a random or quasi-Monte Carlo sample at eachinner node of the Huffman tree to probabilistically select a child nodeto access.

As another example, the data tree may include a kd-tree. In this case,the kd-tree may also be constructed over the partitioned domainutilizing the probability assigned to each of the plurality of sets.Furthermore, the kd-tree may be constructed utilizing a heuristic (e.g.a mid-split predictor, an entropy predictor, or a hybrid predictor,etc.). As another option, the constructing of the kd-tree may includereducing a search space for finding a minimum cost function.

In this way, given an arbitrary-dimensional voxelized domain withprobabilities assigned to each cell, an arbitrary number ofwell-distributed samples may be drawn inside the voxel cells accordingto their probabilities. Both the samples inside voxel cells and samplesover the domain as a whole may be well-distributed. As an option, toselect domain cells with a small number of expected traversal steps, aHuffman tree may be used. In this case, samples inside the domain cellsmay be enumerated using a suited low-discrepancy sequence.

In the context of the present description, a low-discrepancy sequencerefers to any sequence with the property that for all values of N, itssubsequence x₁, . . . ,x_(N) has a low discrepancy. For example, thelow-discrepancy sequence may include a Halton or a Sobol sequence, etc.As another option, to select domain cells with a small number ofexpected computation steps a kd-tree may be used.

More illustrative information will now be set forth regarding variousoptional architectures and features with which the foregoing frameworkmay or may not be implemented, per the desires of the user. It should bestrongly noted that the following information is set forth forillustrative purposes and should not be construed as limiting in anymanner. Any of the following features may be optionally incorporatedwith or without the exclusion of other features described.

FIG. 2 shows a method 200 for enumerating samples using a Huffman tree,in accordance with another embodiment. As an option, the present method200 may be implemented in the context of the functionality of FIG. 1. Ofcourse, however, the method 200 may be carried out in any desiredenvironment. It should also be noted that the aforementioned definitionsmay apply during the present description.

As shown, a Huffman tree is built over a partitioned domain usingprobabilities assigned to sets. See operation 202. After the Huffmantree is built, the Huffman tree is traversed down to a single domainset. See operation 204.

In one embodiment, traversing the Huffman tree down to a single domainset may be accomplished by sample warping. In another embodiment,traversing the Huffman tree down to a single domain set may beaccomplished by drawing a random or quasi random sample at each innernode to probabilistically select the child node to visit next.

Once a set of the domain is reached at the leaf level of the Huffmantree, the next sequence sample is enumerated inside this set. Seeoperation 206. In one embodiment, the next sequence sample may include aHalton sequence sample. In another embodiment, the next sequence samplemay include a Sobol sequence sample.

More information regarding enumerating the sequence sample inside a setmay be found in U.S. patent application Ser. No. 12/241,928, filed Sep.30, 2008, entitled “COMPUTER GRAPHICS WITH ENUMERATING QMC SEQUENCES INVOXELS,” which is incorporated herein by reference in its entirety forall purposes.

With either sequence, distribution properties may be obtained byenumerating the quasi-Monte Carlo points directly inside voxels thatcorrespond to the leaf-level of probability trees. Once the nextsequence sample is enumerated inside the set, it is determined whether asufficient number of samples have been drawn. See operation 208.

If a sufficient number of samples have not been drawn, the tree istraversed again and sequence samples are enumerated until a sufficientnumber of samples have been drawn. In this case, the number of samplesthat are sufficient may be determined by the system enumerating thesamples or an operator thereof.

FIG. 3 shows a method 300 for constructing kd-trees with low expectedtraversal depth and importance sampling for such kd-trees, in accordancewith another embodiment. As an option, the present method 300 may beimplemented in the context of the functionality of FIGS. 1-2. Of course,however, the method 300 may be carried out in any desired environment.Again, the aforementioned definitions may apply during the presentdescription.

In operation, a kd-tree may be built over a partitioned domain using theprobabilities assigned to sets. First, the whole domain region may bepushed onto a stack. See operation 302. It is then determined whetherthe stack is empty. See operation 304.

If it is determined that the stack is not empty, the domain region ispopped from the stack. See operation 306. It is then determined whetherthe domain region is greater than the leaf size. See operation 308.

If the domain size is greater than the leaf size, a split axis and splitplane are found using a heuristic. See operation 310. In this case, asplit-plane may be found at each inner node.

In one embodiment, finding the split-plane at each inner node mayinclude finding a minimum of a cost function inferred from at least oneof a mid-split predictor, an entropy predictor, or a hybrid predictor.In one embodiment, finding the split-plane at each inner node mayinclude a reduction of the search space (e.g. by binning, etc.).

Once the split axis and the split plane are found using the heuristic,an inner node is created and the split axis and split plane are saved.See operation 312. The child domain regions are then pushed onto thestack. See operation 314.

If it is determined, in operation 308, that the domain region is notgreater than the leaf size, a leaf node is created. See operation 316.It is then again determined if the stack is empty.

If the stack is empty, the kd-tree is traversed to obtain a sample. Seeoperation 318. It is then determined whether a sufficient number ofsamples have been obtained. See operation 320.

If a sufficient number of samples have not been drawn, the tree istraversed again until a sufficient number of samples have been drawn. Inthis case, the number of samples that are sufficient may be determinedby the system enumerating the samples or an operator thereof.

In this way, kd-trees may be generated that can be used to samplepartitioned domains. For generating samples corresponding to voxelprobabilities, kd-trees may be generated that have a small number ofexpected traversal steps in the sample generation stage. Furthermore, invarious embodiments, different heuristics may be utilized to generatethe trees and perform the sampling. In one embodiment, an entropy basedheuristic may be utilized.

Additionally, in another embodiment, samples in voxels may be enumeratedbased on quasi-Monte Carlo or low discrepancy sequences. In this case, aHuffman-tree may be utilized to select the voxels, as discussed above.Consequently, the expected number of traversal steps for voxel selectionmay be as low as possible.

In either case, well-distributed sample points may be generated acrossthe partitioned domain, resulting in higher convergence of integralestimations. In one embodiment, such techniques may be utilized in thecontext of image-based lighting in renderers (e.g. ray-tracing basedrenderers, etc.), where environment maps are used as a light source.Further applications may include multidimensional particle maps based onvoxel cells, volume data visualization, and generally the computation ofintegrals of multi-dimensional partitioned domains.

Importance sampling of partitioned domains, where a probability, orproximate probability, is known for each voxel, is often a standardtechnique used for Monte Carlo or quasi-Monte Carlo techniques. Quiteoften, a function is at least in part defined by some discrete energydensity, such as a textured (environment) light source in computergraphics. The sampling speed for importance sampling of discrete domainsmay be optimized by utilizing probability trees and minimizing theexpected traversal depth. In various embodiments, different techniquesmay be utilized to sample from a discrete domain, either based on theprobabilities of the regions or by incorporating spatial connectivitybetween them.

In one embodiment, cumulative distribution functions (CDFs) may beutilized to select from a discrete set of event. For example, if p_(i)denotes the probability of the ith event for i=1, . . . ,n, thecorresponding CDF is constructed inductively as c₀=0 andc_(i):=c_(i−1)+p_(i) for i=1, . . . ,n.

Sampling is then straightforward: a pseudorandom number ξ ε [0, 1) mapsto the ith event, if c_(i−1)≦ξ<c_(i). Additionally, CDFs used incombination with binary search may also be seen as a 1d-tree using amid-split construction and specialized traversal where no rescaling isneeded. Numerical precision of CDFs is limited by the cumulative sumthat may get more and more unstable with increasing number of events. Ifthe events correspond to k-dimensional domains (e.g. voxels in a grid),k new pseudorandom numbers may then be used for stratification insidethe voxel.

In another embodiment, a probability tree may be utilized to sample froma discrete set of events. A probability tree is a tree where at eachnode all children are assigned probabilities. In other words, each nodesums up to one for all children. The probability for a leaf is theproduct of all probabilities encountered on the traversal.

In one embodiment, the optimal probability tree with respect to theexpected number of traversal steps may be constructed using a greedyHuffman code algorithm. The algorithm starts with nodes corresponding tothe probabilities of single events. Afterwards, the two nodes with thelowest probabilities are successively selected and removed from the poolof nodes. A parent node to these nodes is created, which is assigned thesum of the probabilities of the two children.

This parent node is inserted into the pool of nodes. The algorithmterminates when there is only one node left in the pool, which is theroot node of the tree. As an option, the selection of nodes with thelowest probability may be efficiently implemented using a priorityqueue. The pseudo-code for a Huffman tree construction algorithm isshown in Table 1, in accordance with one embodiment.

TABLE 1 Node huffman (double p₁, . . . , double p_(n)) {   // use a heapas priority queue,   // initialize it with leaf nodes   Heap heap =heapify (Node(p₁), . . . , Node(p_(n) ))   while (heap.size ( ) > 1)   {    Node node1 = heap.extract_min ( );     Node node2 = heap.extract_min( );     Node parent;     parent.left = node1;     parent.right = node2;    parent.prob = node1.prob + node2.prob;     heap.insert (parent);   }  Return heap }

The average code word length l (p₁, . . . , p_(n)) of a Huffman code(which corresponds to the expected number of traversal steps) is boundedby H(p₁, . . . , p_(n))≦l(p₁, . . . , p_(n))≦H(p₁, . . . , p_(n))+1,where

${H\left( {p_{1},\ldots\mspace{14mu},p_{n}} \right)}:={\sum\limits_{i = 1}^{n}p_{i}}$log₂ p_(i) is the entropy.

The case where the branching factor of the probability tree is up to b(i.e. each node has up to b children), an analogous result applies. TheHuffman algorithm then selects the b nodes with lowest probabilities.Consequently, the log₂ is to be replaced by a log_(b), decreasing theexpected traversal depth by a constant factor of log₂(b).

A kd-tree is a space partitioning tree for data in k dimensions, whereall the split planes are axis-aligned. As an option, the kd-tree may bea binary tree. At each node of the tree, exactly one dimension may besplit at a specific position along the according axis, which may bestored in the node. In case of a probability kd-tree, this position maycorrespond to the probability of the left child.

In general, kd-trees are a restriction of general probability treeswhere spatial connectivity is preserved. However, this spatialconnectivity may be exploited. One version of a kd-tree construction isa mid-split tree. Using a mid-split tree, the longest axis is split inthe middle. Table 2, shows algorithm for splitting a k-dimensionalregion defined by the minimum x₀=(x₀ ⁽⁰⁾, . . . ,x₀ ^((k−1))) and themaximum x₁=(x₁ ⁽⁰⁾, . . . ,x₁ ^((k−1))) (with x₀ ^((i))<x₁ ^((i)) fori=0, . . . ,k−1) along the axis of longest extent, in accordance withone embodiment.

TABLE 2 void split (x₀ , x₁) {   int Δ[k] = x₁−x₀;   if (Δ == (1, . . .,1))   {     // create a leaf node     process_leaf (x₀);   {   else   {    // find axis of longest extent     int t = argmax Δ^((i))      0≦i<k     int s = (x₀ ^((t)) + x₁ ^((t)))/ 2;     // processchildren recursively     split (x₀,(x₀ ⁽⁰⁾, . . . ,x₁ ^((t−1)),s,x₁^((t+1)), . . . x₁ ^((k−1))));     split ((x₀ ⁽⁰⁾, . . . ,x₀^((t−1)),s,x₀ ^((t+1)), . . . x₀ ^((k−1))),x₁);    } }

For a mid-split kd-tree, the expected traversal depth l(p₁, . . .,p_(n)) is bounded by log₂(n)≦l(p₁, . . . ,p_(n))<log₂(n)+1. It shouldbe noted that for p_(i)=1/n (i=1, . . . ,n), these bounds are equivalentto H(p₁, . . . , p_(n))≦l(p₁, . . . , p_(n))≦H(p₁, . . . , p_(n))+1,described above. Therefore, the resulting probability tree is close tooptimal. If areas have zero probability, these areas may be removed fromtree creation and the corresponding trees are shallower.

Different techniques may be implemented to use a given probability treefor importance sampling. If the leaves in the probability treecorrespond to k-dimensional voxels, at least k pseudorandom numbers maybe used for sampling. Furthermore, the tree may be traversed in aprobabilistic manner. In other words, a child may be chosen according toits probability. In some cases, a straightforward approach of using onepseudorandom number per child decision is not an option, sincegenerating the pseudorandom numbers is expensive and up to d numbers arenecessary, where d denotes the maximum leaf depth of the tree.

Another approach is to use only one pseudorandom number for the completetraversal. For each traversal step, the probabilities p_(l) and p_(r),with p_(l)+p_(r)=1, of the left and right child are used for choosingthe child to traverse. In this case, ξ ε [0, 1) may denote the currentsample. If ξ<p_(l), the left child is chosen and the sample is rescaledto ξ′=ξ/p_(l). Otherwise, the right child may be selected and the samplemay be rescaled to ξ′=(ξ−p_(l))/p_(r).

In the next traversal step, ξ′ ε [0, 1) is used as the pseudorandomsample. Using this scheme, one component of a sample stratified in k+1dimensions can be used for traversal, while the remaining k componentsyield a good stratification inside the voxel. In this case, a certainamount of precision may be lost at each rescaling event.

In order to increase the numerical precision, more than one pseudorandomnumber may be used (e.g. k for a kd-tree, one per dimension, etc.).Then, the split axis of each node may determine which pseudorandomnumber is rescaled.

In some cases, decoupling the voxel sampling from tree traversal mayscramble the mapping from the unit hypercube to the complete domain. Inother words, the mapping may be arbitrary and may not preserve spatialproximity of the samples. Regarding the sampling of an environment map,the stratification properties of the point set apply to a pixel, but notthe environment map as a whole.

In case of classic Monte Carlo sampling, this is generally not an issue,since the randomness stays the same. However, for quasi-Monte Carlosequences this may lead to degraded performance or more artifacts.

In order to be able to exploit the stratification properties of certainpoint sets, those properties may be preserved during mapping the valuesfrom [0, 1)^(k) to the transformed domain represented by the importancesampling scheme. Contrary to general probability trees, kd-treespreserve parts of the spatial connectivity, which may be exploitedduring sampling. One popular algorithm to traverse kd-trees forimportance sampling while still keeping stratification properties of thepoints is sample warping.

A sample warping algorithm uses the original point set for bothtraversing and stratification in the voxel by smart rescaling. Oneexample of a sample warping algorithm for a binary kd-tree is shown inTable 3.

TABLE 3 void warp_sample (double ξ[k], Node n) {   // traverse tree  while (n is not leaf)   {     int axis = n.axis;     double ξ_(s) = ξ[axis];     if (ξ_(s) < node.p₁)     {       ξ_(s) /= node.p₁;       n =n.left;     }     else     }       ξ_(s) = (ξ_(s) − node.p₁) / (1.0 −node.p₁);       n = n.right;     }     ξ [axis] = ξ_(s);   }   // setfinal position in leaf   for (int i = 0 ; i < k; ++i)      ξ [i] =node.min [i] + node.size [i]* ξ [i]; }

The algorithm outlined in Table 3 is implemented such that positions fora leaf are stored. In another embodiment, the inner nodes may store theposition for the split axis only and that could be used to directlytransform to the desired range.

However, if k>1, the mapping may be discontinuous and some usefulproperties (e.g. minimum distance, etc.) may be lost. Furthermore, therescaling process may suffer from severe bit-loss if the tree is deep.In general, the algorithms perform well for common resolutionvoxelizations and are useful for progressive quasi-Monte Carlointegration.

In some cases, very good distribution properties may be obtained byenumerating the quasi-Monte Carlo points or low discrepancy sequencesdirectly inside voxels that correspond to the leaf-level of probabilitytrees. In one embodiment, the Halton or Sobol low discrepancy sequencealgorithms may be utilized for this task (including all previoussampling tasks described thus far).

First, a voxel may be chosen according to its probability. This ispossible using stratified random sampling over the traversal dimensions.After the voxel has been selected, the next quasi-Monte Carlo point forthe voxel is generated. This may require storing the number ofpreviously generated samples inside each voxel.

The result of this scheme may be compared with generating a very densequasi-Monte Carlo sample set over the whole domain and then selectivelythinning out regions according to their probability, similar torejection sampling. This algorithm may be utilized on any probabilitytree and not only kd-trees. For example, the algorithm may be utilizedon the Huffman tree, which in some cases may be optimal regarding thenumber of expected traversal steps. This is possible as the algorithmdoes not rely on spatial connectivity of tree node regions. Therefore,selecting the voxel where to generate the next sample is as fast astheoretically possible.

However, in order to stratify over the traversal dimensions, the numberof samples or the number of samples for each pass should be known inadvance. Additionally, in some cases, it may be hard to integrate thisscheme into a full-featured quasi-Monte Carlo estimator that needs togenerate additional sample components. Since sampling a domain mightonly represent a small part of the whole system, certain instances (i.e.indices of the quasi-Monte Carlo or low discrepancy sequence) might havebeen used before.

When a different instance is then used for enumerating the next sampleinside the voxel, correlation artifacts may appear. Furthermore, it maybe unclear which instance should be used to generate additional samplecomponents afterwards.

For an arbitrary subtree T, T_(l) may be defined to be the left childsubtree and T_(r) to be the right child subtree, respectively. Theexpected number of traversal steps for T, denoted by E[T], may berecursively written as E[T]=1+p_(l)·E[T_(l)]+p_(r)·E[T_(r)], where p_(l)and p_(r) are the probabilities of the sub-trees.

An exhaustive recursive search over all possible splits via backtrackingmay be possible for very moderate domain sizes and dimensions. Dynamicprogramming allows finding the optimal kd-tree to be solved by using atable to store the expected traversal length of all possible subtrees.Beginning at the leaf level, the expected number of traversal steps maybe evaluated by iterating over all possible split planes and finding theminimum of E[T]=1+p_(l)·E[T_(l)]+p_(r)·E[T_(r)], while looking upE[T_(l)] and E[T_(r)] in the table, since they have already beencomputed.

The memory requirement and computational effort of this technique isO(r₁ ². . . r_(k) ²), where r_(i) is the resolution along dimension i.In some cases, this may only be feasible for small resolutions and smalldimensions. Therefore, the split may be determined based on using onlylocal information derived from domain size and probability.

In one embodiment, it may be assumed that both children are constructedusing a mid-split predictor heuristic. As the value E[T] can beanalytically expressed for a mid-split kd-tree, [log₂ m(T)], where m(T)is the number of leaves in T, it can be computed.

Using these values as a predictor for the real expected traversal depth,a heuristic may be provided to choose a split plane using onlyprobability and domain size, as shown in Expression 1.E[T]≈1+p_(l)·┌ log₂ m(T_(l))┐+p_(r)·┌ log₂ m(T_(r))┐  Expression 1

In some cases, employing the heuristic shown in Expression 1 may yieldslightly better kd-trees than a mid-split heuristic. As another option,a simpler heuristic may be utilized. Expression 2 shows an example ofanother heuristic, in accordance with one embodiment.E[T]≈1+p_(l)·m(T_(l))+p_(r)·m(T_(r))   Expression 2

In another embodiment, a heuristic based on the entropy of the subtreesmay be utilized. For example, if Huffman trees were created for bothchildren, then the expected traversal depth may be estimated as shown inExpression 3, where H(T_(l)) and H(T_(r)) are the entropy of the leavesin the left and right subtree, respectively.E[T]≈1+p_(l)·H(T_(l))+p_(r)·H(T_(r))   Expression 3

Depending on the distribution of probabilities in a subtree theimprovement with respect to expected traversal depth might not be worththe effort. Especially in the case where there is almost equaldistribution in a region, a mid-split may be close to optimal.

The entropy of a region yields a very powerful measure on how promisinga more expensive heuristic may be. A criterion may be applied based oncomparing the maximum entropy ┌ log₂ m(T)┐ with H(T). For example, themore expensive entropy predictor heuristic may be applied if H(T)<c·┌log₂ m(T)┐, where c<1 is a user-controlled threshold factor, and performa simple mid-split otherwise. In this way, effort may be invested whereperformance is most promising. In another embodiment, the optimalkd-tree may be built on a downsampled version of an image, the simplerheuristic may be used on the highest levels of the resulting tree,subsequently switching to the full-resolution input.

To evaluate the probability and entropy for arbitrary regions of thedomain, a summed-area table (SAT) may be used. As an option, such asummed-area table may be constructed rapidly in parallel using the scanprimitive of current graphic processing units (GPUs).

In most cases, the entropy is defined for a set of probabilities p_(i),with Σ_(i)p_(i)=1. For arbitrary regions of the domain, theprobabilities may be renormalized using the constant factors:=1/Σ_(i)p_(i). The SAT may be used for the entropy values since:

$\begin{matrix}{H = {- {\sum\limits_{i}{\left( {p_{i} \cdot s} \right) \cdot {\log_{2}\left( {p_{i} \cdot s} \right)}}}}} \\{= {{- s}{\sum\limits_{i}{p_{i} \cdot \left( {{\log_{2}p_{i}} + {\log_{2}s}} \right)}}}} \\{= {{{- s}{\sum\limits_{i}{{p_{i} \cdot \log_{2}}p_{i}}}} - {{s \cdot \log_{2}}s{\sum\limits_{i}p_{i}}}}} \\{= {{{- s}{\sum\limits_{i}{{p_{i} \cdot \log_{2}}p_{i}}}} - {\log_{2}{s.}}}}\end{matrix}$

Both the values 1/s=Σ_(i)p_(i) and Σ_(i)p_(i)·log₂ p_(i) can beretrieved from their respective SATs.

If one sample number is used for all traversal steps by rescaling, theprecision loss should be taken into account. As an example, aprobability midsplit in a binary tree may be considered, where bothchildren have a probability of 0.5 at all nodes. A traversal with onepseudo-random number may lead to a bit loss of exactly one bit pertraversal step.

A common environment map, for example, may have a resolution of1024×1024, leading to a tree of a depth of about 20. In this scenario,at least 20 bits may be lost during traversal, essentially reducing adouble precision to a single precision floating point number.

Similar precision issues have to be taken into account if the samplewarping algorithm is employed. In that scenario, one sample number maybe used for each dimension of the kd-tree and the bits left over fromtraversing may be used to place the sample into the leaf, reducing theprecision in a leaf. Of course, in case of an additional transformationin place to map from the voxels to the real domain (e.g. forlongitude/latitude environment maps, etc.), the prediction of bit-lossmay be more complicated.

However, double precision floating point may be precise enough for thesample warping algorithm to be successfully employed. Reducing thenumber of expected traversal steps may reduce the average bit loss.

It should be noted that, in one embodiment, a row/column SAT may beemployed. Using a row/column SAT, look-up operations may be saved perprobability/entropy evaluation, since each axis is traversed during thefull entropy predictor heuristic computation.

Additionally, because the leaf probabilities are usually small, andexact values are not depended upon, a fast, approximate log₂implementation for floating-point numbers, may be utilized.

The inner nodes may be rescaled in a way that only a single “splitplane” is stored and the decision for the next child simplifies to aless than comparison, analogous to CDFs. Because the interval ofpossible samples for each node is known, the split plane inside thisinterval may be computed according to the probabilities of the children,as shown by Expression 4, where s denotes the split position, [I₀, I₁)denotes the interval of possible samples at the current node, and p_(l)and p_(r) denote the (unnormalized) probabilities of the children.

$\begin{matrix}{s:={I_{0} + {\left( {I_{1} - I_{0}} \right) \cdot \frac{p_{I}}{p_{I} + p_{r}}}}} & {{Expression}\mspace{20mu} 4}\end{matrix}$

Each leaf may then store the I₀ and the factor 1/(I₁−I₀) to correctlyrescale the sample to [0, 1). In this way, there is only a singlesubtraction and multiplication per sample dimension for a fulltraversal.

As another option, monotony in the cost function may be assumed along anaxis. Although this may not be the case, a local minimum may not beworse than a global minimum. Starting at the mid-split and thensuccessively going to the left or right until the cost functions startsto rise, and then taking the minimum found may allow for the minimumsearch to be aborted the minimum earlier. Still yet, in one embodiment,binning may be used to reduce the number of cost function evaluations.

FIG. 4 shows a probability tree 400, where inner nodes hold the sum ofthe probabilities of their children, in accordance with one embodiment.FIG. 5 shows a decision tree 500 corresponding to FIG. 4 that may beused for sample warping, in accordance with one embodiment. FIG. 6 showsa tree 600 of sample intervals for each node corresponding to FIG. 5,when 1d-sample warping is used, in accordance with one embodiment. Forthe fast sample warping technique, the split planes may be set to thecenter of these intervals.

FIG. 7 shows a table 700 illustrating statistics for different probes,in accordance with one embodiment. As an option, the table 700 may beviewed in the context of the functionality and architecture of FIGS.1-6. Of course, however, the table 700 may be viewed in the context ofany desired environment. It should also be noted that the aforementioneddefinitions may apply during the present description.

The table 700 shows statistics for different probes of size 64×32. Inthis example, the random image utilized is made of random uniformpixels, while the starfield image includes some random pixels that aremuch brighter than the background. For a comparison, the table 700 showsstatistics employing various tree constriction algorithms for a seriesof high dynamic range images as for image based lighting. As an option,3d-trees may be used when an environment map is a HDR video and theshutter time of a single rendered image spans over multiple frames inthe video. Environment rays would then feature time instants where theillumination contributes most.

FIG. 8 illustrates an exemplary system 800 in which the variousarchitecture and/or functionality of the various previous embodimentsmay be implemented. As shown, a system 800 is provided including atleast one host processor 801 which is connected to a communication bus802. The system 800 also includes a main memory 804. Control logic(software) and data are stored in the main memory 804 which may take theform of random access memory (RAM).

The system 800 also includes a graphics processor 806 and a display 808,i.e. a computer monitor. In one embodiment, the graphics processor 806may include a plurality of shader modules, a rasterization module, etc.Each of the foregoing modules may even be situated on a singlesemiconductor platform to form a graphics processing unit (GPU).

In the present description, a single semiconductor platform may refer toa sole unitary semiconductor-based integrated circuit or chip. It shouldbe noted that the term single semiconductor platform may also refer tomulti-chip modules with increased connectivity which simulate on-chipoperation, and make-substantial improvements over utilizing aconventional central processing unit (CPU) and bus implementation. Ofcourse, the various modules may also be situated separately or invarious combinations of semiconductor platforms per the desires of theuser.

The system 800 may also include a secondary storage 810. The secondarystorage 810 includes, for example, a hard disk drive and/or a removablestorage drive, representing a floppy disk drive, a magnetic tape drive,a compact disk drive, etc. The removable storage drive reads from and/orwrites to a removable storage unit in a well known manner.

Computer programs, or computer control logic algorithms, may be storedin the main memory 804 and/or the secondary storage 810. Such computerprograms, when executed, enable the system 800 to perform variousfunctions. Memory 804, storage 810 and/or any other storage are possibleexamples of computer-readable media.

In one embodiment, the architecture and/or functionality of the variousprevious figures may be implemented in the context of the host processor801, graphics processor 806, an integrated circuit (not shown) that iscapable of at least a portion of the capabilities of both the hostprocessor 801 and the graphics processor 806, a chipset (i.e. a group ofintegrated circuits designed to work and sold as a unit for performingrelated functions, etc.), and/or any other integrated circuit for thatmatter.

Still yet, the architecture and/or functionality of the various previousfigures may be implemented in the context of a general computer system,a circuit board system, a game console system dedicated forentertainment purposes, an application-specific system, and/or any otherdesired system. For example, the system 800 may take the form of adesktop computer, lap-top computer, and/or any other type of logic.Still yet, the system 800 may take the form of various other devicesincluding, but not limited to, a personal digital assistant (PDA)device, a mobile phone device, a television, etc.

Further, while not shown, the system 800 may be coupled to a network[e.g. a telecommunications network, local area network (LAN), wirelessnetwork, wide area network (WAN) such as the Internet, peer-to-peernetwork, cable network, etc.] for communication purposes.

While various embodiments have been described above, it should beunderstood that they have been presented by way of example only, and notlimitation. Thus, the breadth and scope of a preferred embodiment shouldnot be limited by any of the above-described exemplary embodiments, butshould be defined only in accordance with the following claims and theirequivalents.

What is claimed is:
 1. A method, comprising: partitioning a domain,utilizing a processor, into a plurality of sets; assigning a probabilityto each of the plurality of sets; constructing a tree over thepartitioned domain utilizing the probability assigned to each of theplurality of sets; selecting a subtree from the tree, the selectedsubtree having a left subtree and a right subtree; calculating anentropy of at least one leaf of the left subtree of the selectedsubtree; calculating an entropy of at least one leaf of the rightsubtree of the selected subtree; calculating an expected number oftraversal steps for the selected subtree as a function of a probabilityof the left subtree, a probability of the right subtree, the entropy ofthe at least one leaf of the right subtree, and the entropy of the atleast one leaf of the left subtree; splitting the selected subtree intoat least two portions based on the calculated expected number oftraversal steps, wherein the selected subtree is split into the at leasttwo portions in response to a determination that a total entropy is lessthan a predetermined maximum entropy, the total entropy including a sumof an entropy of all of the leaves of the left subtree of the selectedsubtree and an entropy of all of the leaves of the right subtree of theselected subtree; and traversing at least one of the at least twoportions of the selected subtree by generating samples from theplurality of sets, the samples being generated according to theprobability of a corresponding set.
 2. The method of claim 1, whereinthe samples are generated utilizing the tree.
 3. The method of claim 2,wherein the tree includes a Huffman tree.
 4. The method of claim 3,further comprising enumerating the samples using a low discrepancysequence.
 5. The method of claim 2, wherein the tree includes a kd-tree.6. The method of claim 5, wherein the kd-tree is constructed over thepartitioned domain utilizing the probability assigned to each of theplurality of sets.
 7. The method of claim 6, wherein the kd-tree isconstructed utilizing a heuristic.
 8. The method of claim 1, wherein thecalculation of the expected number of traversal steps for the selectedsubtree includes calculating a product of the probability of the leftsubtree and the entropy of all of the leaves of the left subtree, summedwith a product of the probability of the right subtree and the entropyof all of the leaves of the right subtree.
 9. A computer program productembodied on a non-transitory computer readable medium, comprising:computer code for partitioning a domain into a plurality of sets;computer code for assigning a probability to each of the plurality ofsets; computer code for constructing a tree over the partitioned domainutilizing the probability assigned to each of the plurality of sets;computer code for selecting a subtree from the tree, the selectedsubtree having a left subtree and a right subtree; computer code forcalculating an entropy of at least one leaf of the left subtree of theselected subtree; computer code for calculating an entropy of at leastone leaf of the right subtree of the selected subtree; computer code forcalculating an expected number of traversal steps for the selectedsubtree as a function of a probability of the left subtree, aprobability of the right subtree, the entropy of the at least one leafof the right subtree, and the entropy of the at least one leaf of theleft subtree; computer code for splitting the selected subtree into atleast two portions based on the calculated expected number of traversalsteps, wherein the computer program product is operable such that theselected subtree is split into the at least two portions in response toa determination that a total entropy is less than a predeterminedmaximum entropy, the total entropy including a sum of an entropy of allof the leaves of the left subtree of the selected subtree and an entropyof all of the leaves of the right subtree of the selected subtree; andcomputer code for traversing at least one of the at least two partitionsof the selected subtree by generating samples from the plurality ofsets, the samples being generated according to the probability of acorresponding set.
 10. An apparatus, comprising: a processor for:partitioning a domain into a plurality of sets, assigning a probabilityto each of the plurality of sets, constructing a tree over thepartitioned domain utilizing the probability assigned to each of theplurality of sets, selecting a subtree from the tree, the selectedsubtree having a left subtree and a right subtree, calculating anentropy of at least one leaf of the left subtree of the selectedsubtree, calculating an entropy of at least one leaf of the rightsubtree of the selected subtree, calculating an expected number oftraversal steps for the selected subtree as a function of a probabilityof the left subtree, a probability of the right subtree, the entropy ofthe at least one leaf of the right subtree, and the entropy of the atleast one leaf of the left subtree, splitting the selected subtree intoat least two portions based on the calculated expected number oftraversal steps, wherein the processor is operable such that theselected subtree is split into the at least two portions in response toa determination that a total entropy is less than a predeterminedmaximum entropy, the total entropy including a sum of an entropy of allof the leaves of the left subtree of the selected subtree and an entropyof all of the leaves of the right subtree of the selected subtree, andtraversing at least one of the at least two portions of the selectedsubtree by generating samples from the plurality of sets, the samplesbeing generated according to the probability of a corresponding set. 11.The computer program product of claim 9, wherein the samples aregenerated utilizing the tree.
 12. The computer program product of claim11, wherein the tree includes a Huffman tree.
 13. The computer programproduct of claim 12, further comprising enumerating the samples using alow discrepancy sequence.
 14. The computer program product of claim 11,wherein the tree includes a kd-tree.
 15. The computer program product ofclaim 14, wherein the kd-tree is constructed over the partitioned domainutilizing the probability assigned to each of the plurality of sets. 16.The computer program product of claim 15, wherein the kd-tree isconstructed utilizing a heuristic.
 17. The apparatus of claim 10,wherein the samples are generated utilizing the tree.
 18. The apparatusof claim 17, wherein the tree includes a Huffman tree.
 19. The apparatusof claim 18, further comprising enumerating the samples using a lowdiscrepancy sequence.
 20. The apparatus of claim 17, wherein the treeincludes a kd-tree.
 21. The apparatus of claim 20, wherein the kd-treeis constructed over the partitioned domain utilizing the probabilityassigned to each of the plurality of sets.